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Mirrors > Home > MPE Home > Th. List > notzfausOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of notzfaus 5227 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
notzfaus.1 | ⊢ 𝐴 = {∅} |
notzfaus.2 | ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) |
Ref | Expression |
---|---|
notzfausOLD | ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notzfaus.1 | . . . . . 6 ⊢ 𝐴 = {∅} | |
2 | 0ex 5175 | . . . . . . 7 ⊢ ∅ ∈ V | |
3 | 2 | snnz 4672 | . . . . . 6 ⊢ {∅} ≠ ∅ |
4 | 1, 3 | eqnetri 3057 | . . . . 5 ⊢ 𝐴 ≠ ∅ |
5 | n0 4260 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
6 | 4, 5 | mpbi 233 | . . . 4 ⊢ ∃𝑥 𝑥 ∈ 𝐴 |
7 | biimt 364 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦))) | |
8 | iman 405 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦)) | |
9 | notzfaus.2 | . . . . . . . 8 ⊢ (𝜑 ↔ ¬ 𝑥 ∈ 𝑦) | |
10 | 9 | anbi2i 625 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦)) |
11 | 8, 10 | xchbinxr 338 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑦) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
12 | 7, 11 | syl6bb 290 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
13 | xor3 387 | . . . . 5 ⊢ (¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝑦 ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
14 | 12, 13 | sylibr 237 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
15 | 6, 14 | eximii 1838 | . . 3 ⊢ ∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
16 | exnal 1828 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
17 | 15, 16 | mpbi 233 | . 2 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
18 | 17 | nex 1802 | 1 ⊢ ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-nul 4244 df-sn 4526 |
This theorem is referenced by: (None) |
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